Osculating spaces to secant varieties

arXiv:math/0406322v1 [math.AG] 16 Jun 2004

Osculating spaces to secant varieties E. Ballico, C. Bocci, E. Carlini, C. Fontanari∗

Abstract We generalize the classical Terracini’s Lemma to higher order osculating spaces to secant varieties. As an application, we address with the so-called Horace method the case of the d-Veronese embedding of the projective 3-space.

A.M.S. Math. Subject Classification (2000): 14N05. Keywords: osculating space, secant variety, Terracini’s Lemma, Laplace equation, Horace method.

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Introduction

Let X ⊂ Pr be an integral non-degenerate projective variety of dimension n. The h-secant variety S h (X) of X is the Zariski closure of the set of points in Pr lying in the span of h + 1 independent points of X. The variety X is said to be h-defective if S h (X) has dimension strictly less than the expected min{(h + 1)(n + 1) − 1, r}. The dimension of S h (X) can be computed via the classical Terracini’s Lemma ([7]), asserting that, for p0 , . . . , ph general points on X and p general in their span, one has Tp (S h (X)) =< Tp0 (X), . . . , Tph (X) > . Let m be a non-negative integer and let Tpm (X) denote the m-osculating space to X at p (in particular, notice that Tp0 (X) = {p} and Tp1 (X) = Tp (X), the usual tangent space). Here we are going to prove the following generalization of Terracini’s Lemma (see [1] for a completely different one, computing the tangent space to the secant variety of an osculating variety): ∗ This research is part of the T.A.S.C.A. project of I.N.d.A.M., supported by P.A.T. (Trento) and M.I.U.R. (Italy).

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Theorem 1. Fix non-negative integers h and m, let p0 , . . . , ph be general ponts on X and let p ∈ S h (X) be general in < p0 , . . . , ph >. Then Tpm (S h (X)) =< Tpm0 (X), . . . , Tpmh (X) > . In particular, it turns out that Tpm (S h (X)) has dimension much less than the expected one (in classical terminology, S h (X) satisfies many differential equations of order m at a general point p). However, this fact is not so surprising: as pointed out in the introductions of [3] and [4], a variety containing a linear space always satisfies several differential equations. Theorem 1 is indeed a consequence of a more general result (see Proposition 1) to be proved in Section 2, where the connections with Terracini’s results on varieties satisfying many differential equations ([8]) are also discussed. As an application of Theorem 1, we can determine the dimension of the osculating spaces to the secant varieties of some remarkable projective varieties. We point out the following straightforward consequences: Corollary 1. (Ciliberto, Cioffi, Miranda, Orecchia [2]) For any integer d ≥ 1, let V2,d denote the d-Veronese embedding of the projective plane P2 . Fix integers h ≥ 1 and 1 ≤ m ≤ 20; if h = 1, assume either d < m or d > 2m−2; if h = 2, assume either d < 3m/2 or d > 2m − 2; if h = 4, assume either d < 2m or d > (5m−2)/2; if h = 5, assume either d < 12m/5 or d > (5m−2)/2; if h = 6, assume either d < 21m/8 or d > (8m − 2)/3; if h = 7, assume either d < 48m/17 or d > (17m − 2)/6. Then the homogeneous linear system OP2 (d)(−mp0 . . . − mph ) is non-special. In particular, we have m+1 m−1 h − 1. dim Tp (S (V2,d )) = (h + 1) 2 Corollary 2. (Laface [6]) For any integer n ≥ 0, let Fn be a Hirzebruch surface, i.e. Fn = P(OP1 ⊗ OP1 (n)), and let F , H be the two generators of Pic(Fn ) such that F 2 = 0, H 2 = n, F.H = 1. If m ≤ 3 and (n, a, b, m, h + 1) ∈ / {(1, 0, 4, 2, 5), (1, 0, 6, 3, 5), (5, 1, 4, 3, 10), (6, 0, 4, 3, 11), (n, 2e, 2, 2, 2e + n + 1), (n, e, 0, 2, r), (n, 4e + n + 1, 2, 3, 2e + n + 1), (n, 3e + 1, 3, 3, 2e + n + 1), (n, 3e, 3, 3, 2e + n + 1), (n, e, 1, 3, r), (n, e, 0, 3, r)}, then the homogeneous linear system OFn (aH + bF )(−mp0 . . . − mph ) is non-special. In particular, we have m+1 m−1 h − 1. dim Tp (S (Fn )) = (h + 1) 2 We are going to prove also the following result: Theorem 2. Fix integers h ≥ 1, d ≥ 4, and let V3,d denote the d-Veronese embedding of the projective space P3 . If either 10(h + 1) + 5d − 12 ≤ 2

d+3 3

or 10(h + 1) ≥ d+3 + 5d − 12, then the homogeneous linear system 3 OP3 (d)(−3p0 . . . − 3ph ) is non-special. In particular, we have

dim Tp2 (S h (V3,d )) = 10(h + 1) − 1. Theorem 2 is only a special case of a couple of results concerning the postulation of double and triple general points in the projective space P3 (see Proposition 3 and Proposition 4) obtained in Section 3 via the so-called Horace method for zero-dimensional schemes (see Lemma 1). We work over the complex field C. We are grateful to Stephanie Yang for providing us with the complete list of special systems of fat points with multeplicity at most 3 in the projective plane, which is certainly well-known but seems to be written nowhere.

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Terracini’s Lemma and Laplace equations

Let X0 , . . . , Xh ⊆ Pr be integral non-degenerate projective varieties of dimension ni and let J = J(X0 , . . . , Xh ) :=

[

< p0 , . . . , p h >

pi ∈Xi

be the join variety of X0 , . . . , Xh . Recall that if p is a smooth point of an integral variety X ⊂ Pr of dimension n and U ⊆ Cn −→ Cr+1 \ {0} t 7−→ p(t) is the lifting of a local parametrization of X centered in p, then the mosculating space Tpm (X) is the projective subspace spanned by the points [pI (0)] ∈ Pr , where I = (i1 , . . . , in ) is a multi-index such that |I| ≤ m and |I| pI = i∂1 p in . With this notation, the following holds: ∂t1 ...∂tn

Proposition 1. Let pi ∈ Xi , i = 0, . . . , h, be general points and let p ∈ J be general in < p0 , . . . , ph >. We have Tpm (J) =< Tpm0 (X1 ), . . . , Tpmh (Xh ) > . Proof. We write the proof in the h = 1 case, the generalization to arbitrary h being obvious. Let Ui ⊆ Xi be an analytic neighborhood of pi and denote 3

by Φi : Ani → Pr a local parameterization of Xi . With this notation, a parameterization of J(X1 , X2 ) in a neighborhood of p is given by: Φ:

An1 × An2 × A2 −→ X q˜ = (˜ q1 , q˜2 , α1 , α2 ) 7→ α1 Φ1 (˜ q1 ) + α2 Φ2 (˜ q2 ).

Since Tpmi (Xi ) is spanned by the derivatives of order less or equal than m of m Φi evaluated in Φ−1 i (pi ) and Tp (J(X1 , X2 )) is spanned by the derivatives of order less or equal than m of Φ evaluated in Φ−1 (p), the result follows. It is clear that Theorem 1 is just the special case of Proposition 1 corresponding to X0 = . . . = Xh = X. Now we fix our attention on differential equations of second order (the so-called Laplace equations) . In [8] Terracini gives a characterization of varieties satisfying K +m T = 2 of such equations, where K is the dimension of the variety and m > 0 (a modern reference is [5], Theorem 1.5). When K grows up, T becomes a huge number and we can ask, for instance, if secant varieties satisfy such a number of Laplace equations. We will see that the answer is negative, provided that we consider varieties without unexpected properties. Namely, let X ⊂ Pr be a variety of dimension n such that X is not k-defective for all k. Thus, by Theorem 1 we have n+2 2 h − 1. (1) dim(Tu (S (X))) ≤ (h + 1) n The previous inequality is strict when the 2-osculating spaces at the generic points of X intersect along subspaces of dimension greater than or equal to zero or they have dimension less than the expected one. We assume that equality holds in (1) and let K := dim(Sech (X)) = (h + 1)n + h. Thus, by Theorem 1, S h (X) satisfies 1 n+2 K +2 n2 h 1 1 = n2 h2 + − (h + 1) T := + nh2 + nh + h2 + h n K 2 2 2 2 Laplace equations. It is easy to see that we can write T as 1 2 1 1 1 K − n − n − 1 h − n2 + n. T = 2 2 2 2 2

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Hence if n ≥ 2 then T ≤ instead n = 1 we have

K 2

and we are not in the hypothesis of [8]. If K + h. T ≥ 2

Hence in this case we are in the hypothesis of [8] and in the notation of [5] we have: Proposition 2. Let C be an integral non-degenerate curve in Pr and suppose that 2h+1 ≤ r, h > 0. Then S h (C) is contained in a variety Uq = ∞t Pp , such that the Pq tangent at the points of a Pp lie in a P2K−t−m , with 0 ≤ t ≤ K −m. Recall that an integral non-degenerate curve C is not k-defective for any k (in particular, S 1 (C) is a 3-fold) and the osculating spaces to C at a general point have always the expected dimension. If C is embedded as a rational normal curve in a projective space Pr with r big enough, then two osculating spaces TP2 (C) and TQ2 (C) do not intersect and S 1 (C) satisfies exactly four Laplace equations. We point out that the case in which X is a 3−fold satisfying four Laplace equations is described in [8] by Terracini. He asserts that X lies in P5 or it is a ∞2 of developable lines (with the tangent P3 fixed along a line) or it lies in an ordinary developable 4−fold.

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The Horace method and postulation of points in P3

For any closed subscheme Z ⊂ Pn and every integer t, let ρZ,t,n : H 0 (Pn , OPn (t)) → H 0(Z, OZ (t)) be the restriction map. Z is said to have maximal rank if for every integer t > 0 the restriction map ρZ,t,n has maximal rank, i.e. it is either injective or surjective. Set ρZ,t := ρZ,t,3 . Let W be an algebraic scheme and H ⊂ W an effective Cartier divisor of W . For any closed subscheme A of W let ResH (A) denote the residual scheme of A with respect to H. Assume now that W projective and fix L ∈ Pic(W ). By the very definition of residual scheme we have the following exact sequence on W : 0 → IResH (A) ⊗ L(−H) → IA ⊗ L → IA∩H,H ⊗ (L|H ) → 0

(2)

From the long cohomology exact sequence of the exact sequence (2) we get the following elementary form of the so-called Horace Lemma: Lemma 1. Let W be a projective scheme, H ⊂ W an effective Cartier divisor of W and A ⊆ W a closed subscheme of W . Then: 5

(a) h1 (W, IA ⊗ L) ≤ h1 (W, IResH (A) ⊗ L(−H)) + h1 (H, IA∩H,H ⊗ (L|H )); (b) h1 (W, IA ⊗ L) ≤ h1 (W, IResH (A) ⊗ L(−H)) + h1 (H, IA∩H,H ⊗ (L|H )). Now assume h1 (W, L) = h1 (W, L(−H)) = h1 (H, L|H ) = 0 and that A is zerodimensional. If the restriction maps H 0 (W, L(−H)) → H 0 (A, L(−H)|ResH (A) ) and H 0 (H, L|H ) → H 0 (A, L|A∩H ) are surjective (resp. injective), then the restriction map H 0 (W, L) → H 0 (A, L|A ) is surjective (resp. injective). Remark 1. Fix integers k ≥ 3, a ≥ 0 and b ≥ 0. Assume (k, a, b) ∈ / {(2, 0, 2), (3, 2, 0), (3, 1, 1), (4, 0, 5), (4, 2, 1), (4, 2, 0), (5, 2, 3), (6, 5, 0), (6, 4, 1)}. Let Z ⊂ P2 be a general union of a triple points and b double points. By the classification of all special linear systems of fat points with multiplicity at most 3 in the projective plane, we obtain that the restriction map ρZ,k,2 : H 0 (P2 , OP2 (k)) → H 0 (Z, OZ (k)) has maximal rank. For all integers t ≥ 2 and n ≥ 2 define the integers at,n and bt,n by the relations: n+2 t+n n+2 (3) , 0 ≤ bt,n < at,n + bt,n = 2 n 2 Set at := at,3 and bt := bt,3 . Notation 1. For all integers t ≥ 2, a ≥ 0 and b ≥ 0 such that t+3 10a + 4b ≤ 3

(4)

we will say that the assertion At,a,b is true if for a general union Z ⊂ P3 of a triple points and b double points the restriction map ρZ,t is surjective. Notation 2. For all integers t ≥ 2, a ≥ 0 and b ≥ 0 such that t+3 10a + 4b ≥ 3

(5)

we will say that the assertion Bt,a,b is true if for a general union Z ⊂ P3 of a triple points and b double points the restriction map ρZ,t is injective. Remark 2. Let Z ⊂ W be zero-dimensional schemes. Since W is affine, the restriction map H 0 (W, OW ) → H 0 (Z, OZ ) is surjective. Hence if At,a,b is true, then At,x,y is true for all pairs of non-negative integers (x, y) such that x ≤ a and x + y ≤ a + b. Obviously, if W ⊂ P3 and h0 (P3 , IZ (t)) = 0, then h0 (P3 , IW (t)) = 0. Hence if Bt,a,b is true, then Bt,u,v is true for all pairs (u, v) of non-negative integers such that u ≥ a and u + v ≥ a + b. 6

Remark 3. Let T be any integral projective variety and V any linear system on T . For any integer s > 0 and general P1 , . . . , Ps ∈ T we have dim(V (−P1 . . . − Ps )) = min{0, dim(V ) − s}. Proposition 3. Fix integers k ≥ 4, a ≥ 0 and b ≥ 0. Set γ := max{0, k − b − 4}. If 10a + 4b + 3γ + 2k ≤ k+3 , then Ak,a,b is true. 3

3 Proof. It is sufficient to show the existence of a disjoint union E ⊂ P of k+3 a triple points and b double points and 3 − 10a − 4b points such that h1 (P3 , IA (k)) = 0, i.e. such that h0 (P3 , IA (k)) = 0. First assume a ≥ ak,2 . Fix a plane H ⊂ P3 . Let S be a general subset of H with card(S) = ak,2 and let E ⊂ P3 be the union of the triple points of P3 such that Ered = S. Hence E ∩ H is the union of ak,2 triple points of H. By its very definition we have 0 ≤ bk,2 ≤ 5. First assume 0 ≤ bk,2 ≤ 2. Let M be a general subset of H with cardinality bk,2 . By Remark 1 we have h1 (H, I(E∩H)∪M (k)) = h0 (H, I(E∩H)∪M (k)) = 0. The residual scheme ResH (E∪M) is the union E ′ of ak,2 double points supported on Ered and bk,2 simple points. Hence by Horace 3 Lemma 1 and Remark 3 it is sufficient to show the existence of a union B ⊂ P k+3 of a−ak,2 triple points, b double points and 3 −10a−4b−bk,2 simple points such that h1 (P3 , IB (k − 1)) = 0, i.e. such that h0 (P3 , IB (k − 1)) = 0. The idea is to use the same strategy with the integer k −1 instead of the integer k. In order to do so, we need to look first at the case 3 ≤ bk,2 ≤ 5. If b = 0 (i.e. γ = k − 4), then we do exactly the same construction. Assume now b > 0. Fix a general P ∈ H. Let M ′ ⊂ H a general subset of H with cardinality bk,2 − 3. Set G := E ′ ∪ 2P ∪ M ′ . Hence length(G ∩ H) = k+2 . By Remark 2 0 1 1 we have h (H, IG (k)) = h (H, IG (k)) = 0. We have ResH (G) = E ′ ∪ {P }. Hence we may try to procede as in the case bk,2 ≤ 2 using E ′ ∪ {P } instead of E ′ for the step concerning OP3 (k − 1). We continue in one of the two ways according to the value of bk−1,2 (of course, if b ≥ 2) decreasing each time the degree of the line bundle, until perhaps (say when we need to consider OP3 (c)) we have only a′ < ac,2 triple points left. In this case we insert on H the support of all a′ triple points and use the integer bc,2 + 6(ak,2 − a′ ) instead of the integer bk,2 . If c = k we handle also the case a < ak,2 excluded at the beginning of the proof. The choice of the integer γ and our assumption 10a + 4b + 3γ + 2k ≤ k+3 shows that we can continue for all lower degree 3 line bundles until we exaust in in this way both the triple points and the double points.

The same proof (just changing the surjectivity part with the injectivity part of Horace Lemma 1) gives the following result:

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Proposition 4. Fix integers k ≥ 4, a ≥ 0 and b ≥ 0. Set γ := max{0, k − k+3 b − 4}. If 10a + 4b ≥ 3 + 3γ + 2k, then Bk,a,b is true.

References [1] E. Ballico and C. Fontanari: A Terracini Lemma for osculating spaces with applications to Veronese surfaces. J. Pure Appl. Algebra (to appear). [2] C. Ciliberto, F. Cioffi, R. Miranda, F. Orecchia: Bivariate Hermite Interpolation and Linear Systems of Plane Curves with Base Fat Points, Proceedings ASCM 2003, Lecture Notes Series on Computing 10, World Scientific Publ., Singapore/River Edge, USA (2003), 87–102. [3] R. H. Dye: Osculating hyperplanes and a quartic combinant of the nonsingular model of the Kummer and Weddle surfaces. Math. Proc. Camb. Phil. Soc. 92 (1982), 205–220. [4] R. H. Dye: The extraordinary higher tangent spaces of certain quadric intersections. Proceedings of the Edimburgh Mathematical Society 35 (1992), 437–447. [5] G. Ilardi: Rational varieties satisfying one or more Laplace equations, Ricerche di Matematica 48 (1999), 123-137. [6] A. Laface: On linear systems of curves on rational scrolls. Geom. Dedicata 90 (2002), 127–144. [7] A. Terracini: Sulle Vk per cui la varietà degli Sh (h + 1)-seganti ha dimensione minore dell’ordinario, Rend. Circ. Mat. Palermo 31 (1911), 392-396. [8] A. Terracini: Sulle Vk che rappresentano pi` u di k(k−1) equazioni di 2 Laplace linearmente indipendenti. Rend. Circ. Mat. Palermo 33 (1912), 11–23.

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Edoardo Ballico Università degli Studi di Trento Dipartimento di Matematica Via Sommarive 14 38050 POVO (Trento) Italy e-mail: [email protected] Cristiano Bocci Università degli Studi di Milano Dipartimento di Matematica “F. Enriques” Via Cesare Saldini 50 20133 MILANO Italy e-mail: [email protected] Enrico Carlini Università degli Studi di Pavia Dipartimento di Matematica “F.Casorati” Via Ferrata 1 27100 PAVIA Italy e-mail: [email protected] Claudio Fontanari Università degli Studi di Trento Dipartimento di Matematica Via Sommarive 14 38050 POVO (Trento) Italy e-mail: [email protected]

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